On the other hand, for \(0<\alpha <2\)  and \(\beta >0\) ,
by Theorems 1.5 and 1.6 (p.35) in
[1]}, we can find a constant \(C_0 = C_0(\alpha ,\beta ) > 0\)  such that
\(\vert E_{\alpha , \beta }(-\lambda t^{\alpha })\vert \le \left\lbrace  \begin{array}{rl}& C_0(1+\vert \lambda \vert t^{\alpha })^{\frac{1-\beta }{\alpha }}\exp (\vert \lambda \vert ^{\frac{1}{\alpha }} t)\quad \mbox{if $-\Lambda _0 \le \lambda < 0$},\\& \frac{C_0}{1+\lambda t^{\alpha }} \quad \mbox{if $\lambda \ge 0$}\end{array}\right.\qquad \mathrm {{(4.4)}}\) 
